Live Exercise 2: Replacement Effect and Social Value of Innovation

MSc-level Industrial Organisation course at the University of St Andrews

Solutions

Group exercise (≈20 minutes)

• Work in groups of 2–3.
• Show all intermediate steps — the algebra matters.
• Full market coverage assumed throughout.

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Problem: Replacement effect and social value

A pharmaceutical company has discovered a new manufacturing process (Tech P) that reduces the marginal cost of producing a drug from \(c_0 = 60\) to \(c_1 = 30\). Market demand for the drug is

\[ P(Q) = 100 - Q. \]

Assume Tech P is available exclusively to the one firm that acquires it (patent protection is perfect).

(a) Is the innovation drastic?

Use the monopoly pricing formula to determine whether Tech P is drastic or non-drastic. State the condition and evaluate it numerically.

Recall: a process innovation is drastic if \(P^m(c_1) < c_0\), where \(P^m(c) = \tfrac{A+c}{2}\) under linear demand \(P = A - Q\).

(b) Monopoly and competitive WTP

Compute the maximum willingness to pay (WTP) for Tech P of:

1. a monopolist (not threatened by entry)
2. a competitive innovator — a firm that operates in a competitive market before acquiring the patent, then gains exclusive rights to Tech P

Hint for (2): use your result from (a) to determine whether the competitive innovator limit-prices at \(p = c_0\) after acquiring the patent.

(c) Social planner’s value

For linear demand \(P = A - Q\) with efficient production (price equals marginal cost), total surplus is \(W(c) = \tfrac{(A-c)^2}{2}\).

1. Compute the social planner’s value of the innovation, \(\Delta W = W(c_1) - W(c_0)\).
2. Arrange the three values — \(\Delta\pi^m\), competitive WTP, \(\Delta W\) — in increasing order.
3. State Arrow’s replacement effect in one sentence. Using the numbers from (b), explain what drives the gap between \(\Delta\pi^m\) and the competitive WTP. Why is \(\Delta W\) larger than both?

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Solution

(a) Drastic or non-drastic?

With \(P(Q) = 100 - Q\) and \(c_1 = 30\), the post-innovation monopoly price is

\[ P^m(c_1) = \frac{A + c_1}{2} = \frac{100 + 30}{2} = 65. \]

Since \(P^m(c_1) = 65 > c_0 = 60\), the innovation is non-drastic.

Intuitively, even with the lower cost, the monopolist would price above the competitive benchmark — so the innovation does not completely destroy the competitive fringe.

(b) Monopoly and competitive WTP

Monopoly WTP

Pre-innovation (\(c_0 = 60\)):

\[ \max_Q (100 - Q - 60)Q = \max_Q (40 - Q)Q. \]

FOC: \(40 - 2Q = 0 \Rightarrow Q_0^m = 20\), \(P_0^m = 80\),

\[ \pi_0^m = (80 - 60)\times 20 = 400. \]

Post-innovation (\(c_1 = 30\)):

\[ \max_Q (70 - Q)Q. \quad \text{FOC: } 70 - 2Q = 0 \Rightarrow Q_1^m = 35,\; P_1^m = 65. \]

\[ \pi_1^m = (65 - 30)\times 35 = 1{,}225. \]

Monopoly WTP:

\[ \Delta\pi^m = \pi_1^m - \pi_0^m = 1{,}225 - 400 = \mathbf{825}. \]

Competitive WTP

Since the innovation is non-drastic, the competitive innovator limit-prices at \(p = c_0 = 60\) after acquiring the patent:

\[ Q = 100 - 60 = 40, \qquad \text{profit} = (c_0 - c_1)\times Q = 30\times 40 = \mathbf{1{,}200}. \]

(c) Social planner’s value and the replacement effect

\[ W(c_0) = \frac{(100 - 60)^2}{2} = \frac{1{,}600}{2} = 800, \qquad W(c_1) = \frac{(100 - 30)^2}{2} = \frac{4{,}900}{2} = 2{,}450. \]

\[ \Delta W = 2{,}450 - 800 = \mathbf{1{,}650}. \]

Increasing order:

\[ \underbrace{825}_{\Delta\pi^m} \;<\; \underbrace{1{,}200}_{\text{competitive WTP}} \;<\; \underbrace{1{,}650}_{\Delta W}. \]

(Check: with linear demand, \(\Delta W = 2\,\Delta\pi^m\) always holds — the planner values the output expansion that the monopolist ignores.)

Replacement effect: The monopolist already earns \(\pi_0^m = 400\) before innovating; innovation only adds 825 to its profit rather than yielding the full post-innovation profit of 1,225. The competitive innovator earns zero before the patent, so its incremental gain (1,200) is larger. This is Arrow’s replacement effect: pre-innovation rents reduce the incumbent’s marginal gain from innovation.

Why $W > $ competitive WTP: The social planner also values the consumer surplus gains from the output expansion (\(Q\) rises from 40 to 70 under efficient pricing at \(c_1\)), which no private firm captures. Neither private incentive approaches the social value of the innovation.